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The Thomas Taylor Series #29
A Commentary on the First Book of Euclid's Elements
In Proclus' penetrating exposition of Euclid's methods and principles, the only one of its kind extant, we are afforded a unique vantage point for understanding the structure and strength of the Euclidean System. A primary source for the history and philosophy of mathematics, Proclus' treatise contains much priceless information about the mathematics and mathematicians of the previous seven or eight centuries that has not been preserved elsewhere. This is virtually the only work surviving from antiquity that deals with what we today would call the philosophy of mathematics.
To all the students interested in the logic and history of mathematics and in the relations between philosophy and mathematics in antiquity, this volume will be an invaluable resource. In his new forward, Ian Muller discusses new scholarship on the commentary and places the work in historical and cultural context.
To all the students interested in the logic and history of mathematics and in the relations between philosophy and mathematics in antiquity, this volume will be an invaluable resource. In his new forward, Ian Muller discusses new scholarship on the commentary and places the work in historical and cultural context.
355 pages, Paperback
Published October 19, 1992
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About the author
Proclus
254 books54 followersProclus Lycaeus (/ˈprɒkləs ˌlaɪˈsiːəs/; 8 February 412 – 17 April 485 AD), called the Successor (Greek Πρόκλος ὁ Διάδοχος, Próklos ho Diádokhos), was a Greek Neoplatonist philosopher, one of the last major Classical philosophers (see Damascius). He set forth one of the most elaborate and fully developed systems of Neoplatonism. He stands near the end of the classical development of philosophy, and was very influential on Western medieval philosophy (Greek and Latin).
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August 2, 2018
I mean wow, this document is very impressive. I just wanted to say I owe it to Proclus and this book to teach me about the finer things of constructing a detailed mathematical document like Euclid’s Elements. The point of the document itself is to give you the very fundamental reasoning WHY the Elements was set up the way it was. I recommend you read Aristotle’s Organon before reading this, or at least his metaphysics, because Proclus really draws on Aristotle here and it is very difficult to follow his esoteric reasoning at times. As someone who was interested in how Euclid and other mathematicians played the ‘fundamental mathematical explanation of the universe’ game, I definitely followed this particular treatise with much interest!
I couldn’t recommend this more to anyone who read Euclid’s Elements and wanted deeper explanations, or if you just want an explanation regarding Neo-Platonism and theoretic arithmetic/geometry in general. Great read 10/10.
I couldn’t recommend this more to anyone who read Euclid’s Elements and wanted deeper explanations, or if you just want an explanation regarding Neo-Platonism and theoretic arithmetic/geometry in general. Great read 10/10.
February 2, 2022
For the most part, what has passed under the name of philosophy of mathematics over the past century or so has been dominated by the hidebound preoccupations of the Anglo-American analytic school and, therefore, can be of but marginal interest to a man of wide culture. Yet, this was not always so. During the heyday of the discovery of the calculus in early modernity, for instance, philosophical issues in mathematics centering on the nature of number, infinitesimals and proper method exercised the great minds of the period and were closely connected with the philosophy of nature, then undergoing revolutionary change. The same holds for the thought of the ancient world to an even greater degree, as we shall discuss in a moment. For, far from being shackled by minutely technical concerns, the ancients saw in mathematics a model of all knowledge and in knowledge the key to wisdom; hence, their philosophizing always serves a revelatory purpose we moderns no longer seek in it, to our loss.
Perhaps the most distinguished and, for the educated man, most congenial representative of the ancient philosophical stance is the Neoplatonist Proclus (fifth century AD), whose commentary on the first book of Euclid’s Elements forms the subject of the present review. Glenn Morrow has done us a great service with his translation and front matter in the edition published by the Princeton University Press. Out of the profusion of stimulating ideas to be found in this major work, to stay within the limitations of space we must set aside a narrow investigation of its technical terminology, rewarding as this would be, and seek to abstract the essential tenor of Proclus’ system. What will appear most foreign to modern sensibilities is his striking moral motivation for the pursuit of mathematics:
We must cultivate that science of geometry which with each theorem lays the basis for a step upward and draws the soul to the higher world, instead of letting it descend among sensibles to satisfy the common needs of mortals and, in aiming at these, neglect to turn away hence. (p. 69)
How, in practice, does this ascent take place?
This, then, is what learning [mathesis] is, recollection of the eternal ideas in the soul; and this is why the study that especially brings us the recollection of these ideas is called the science concerned with learning [mathematike]. Its name thus makes clear what sort of function this science performs. It arouses our innate knowledge, awakens our intellect, purges our understanding, brings to light the concepts that belong essentially to us, takes away forgetfulness and ignorance that we have from birth, sets us free from the bonds of unreason; and all this by the favor of the god who is truly the patron of this science, who brings our intellectual endowments to light, fills everything with divine reason, moves our souls towards Nous, awakens us as it were from our heavy slumber, through our searching turns us back upon ourselves, through our birthpangs perfects us, and through the discovery of pure Nous leads us to the blessed life. (p. 38)
Hence, Proclus, in keeping with Plato’s inscription above the entrance to the Academy [‘Let no one ignorant of geometry enter here’], sees in the pursuit of mathematics an inherently educative function, as the following three quotations describe:
The Timaeus calls mathematical knowledge the way of education, since it has the same relation to knowledge of all things, or first philosophy, as education has to virtue. Education prepares the soul for a complete life through firmly grounded habits, and mathematics makes ready our understanding and our mental vision for turning towards that upper world….For the beauty and order of mathematical discourse, and the abiding and steadfast character of this science, bring us into contact with the intelligible world itself and establish us firmly in the company of things that are always fixed, always resplendent with divine beauty, and ever in the same relationship to one another. (pp. 17-18)
Being thus endowed and led towards perfection, mathematics reaches some of its results by analysis, others by synthesis, expounds some matters by division, others by definition, and some of its discoveries binds fast by demonstration, adapting these methods to its subjects and employing each of them for gaining insight into mediating ideas. Thus its analyses are under the control of dialectic, and its definitions, divisions, and demonstrations are of the same family and unfold in conformity with the way of mathematical understanding. It is reasonable, then, to say that dialectic is the capstone of the mathematical sciences….Nous, then, wraps up the developments of the dialectical methods, binds together from above all the discursiveness of mathematical reasoning, and is the perfect terminus of the upward journey and of the activity of knowing. (pp. 35-36)
This thinking in geometry occurs with the aid of the imagination. Its syntheses and divisions of the figures are imaginary; and its knowing, though on the way to understandable being, still does not reach it, since the understanding is looking at things outside itself. At the same time the understanding sees them by virtue of what it has within; and though employing projections of its ideas, it is moved by itself to make them external. But if it should ever be able to roll up its extensions and figures and view their plurality as a unity without figure, then in turning back to itself it would obtain a superior vision of the partless, unextended, and essential geometrical ideas that constitute its equipment. This achievement would itself be the perfect culmination of geometrical inquiry, truly a gift of Hermes, leading geometry out of Calypso’s arms, so to speak, to more perfect intellectual insight and emancipating it from pictures projected in imagination. (pp. 44-45)
The following quotation illustrates well Proclus’ proclivity for descrying a moral significance in mathematical truths, thus anticipating the typological and symbolist mentality so characteristic of bookish medieval culture:
For these reasons, therefore, they refer right angles to the immaculate essences in the divine orders and their more particular potencies, as causes of the undeviating providence that presides over secondary things—for what is upright, uninclined to evil, and inflexible accords with the character of those high gods—whereas they say that obtuse and acute angles are left in the charge of the divinities that supervise the forthgoing of things and the change and variety of their powers. The obtuse angle is an image of the extension of the forms to everything, while the acute is a likeness of the cause that discriminates and activates all things….Rightly, then, they exhort the soul to make her descent into the world of generation after the undeviating form of the right angle, inclining no more to one side than to the other, nor being affected more by some things than by others, for the possession of fellow-feeling drags her down into the error and indeterminacy of matter. The perpendicular thus is also a symbol of directness, purity, undefiled unswerving force, and all such things, a symbol of divine and intelligent measure….Hence they say that virtue is like rightness, whereas vice is constituted after the fashion of the indeterminate obtuse and acute, possessing both excesses and deficiencies and showing by this more-and-less its own lack of measure. We shall therefore lay it down that the right among rectilinear angles is the image of perfection, undeviating energy, intelligent limit and boundary, and everything similar to them, and that the obtuse and acute angles are likenesses of indefinite change, irrelevant progression, differentiation, partition, and unlimitedness in general. (pp. 106-107)
Hugh of Saint-Victor’s De sacramentis would be an ideal place to draw out the inner connection between Proclus and the medieval sacramental world-view, but that must be deferred to another occasion. For the time being, let us round out our meditation by citing Proclus’ admirable and moving account of beauty:
To those who say these things we can reply by exhibiting the beauty of mathematics on the principles by which Aristotle attempts to persuade us. Three things, he says, are especially conducive to beauty of body or soul: order, symmetry, and definiteness. Ugliness in the body arises from the ascendancy of disorder and from a lack of shapeliness, symmetry, and outline in the material part of our composite nature; ugliness of mind comes from unreason, moving in an irregular and disorderly fashion, out of harmony with reason and unwilling to accept the principles it imposes; beauty, therefore, will reside in the opposites of these, namely, order, symmetry, and definiteness. These characters we find preeminently in mathematical science. We see order in its procedure of explaining the derivative and more complex theorems from the primary and simpler ones; for in mathematics later propositions are always dependent on the predecessors, and some are counted as starting-points, others as deductions from the primary hypotheses. We see symmetry in the accord of the demonstrations with one another and in their common reference back to Nous; for the measure common to all parts of the science is Nous, from which it gets its principles and to which it directs the minds of its students. And we see definiteness in the fixity and certainty of its ideas; for the objects of mathematical knowledge do not appear now in one guise and now in another, like the objects of perception or opinion, but always present themselves as the same, made definite by intelligible forms. If, then, these are the factors especially productive of beauty, and mathematics is characterized by them, it is clear that there is beauty in it. How could it be otherwise when Nous illumines this science from above and its earnest endeavor is to spur us to move from the sense world into that intelligible region? (pp. 22-23)
Certainly, contemporary practicing mathematicians have and wield an informal concept of beauty or elegance, but, as an index of ever-declining educational standards, it seems quite possible indeed that nobody in the past millenium and even longer (since Boethius and Scotus Eriugena, say) has devoted any rigorous reflection to what mathematical beauty truly consists in. It plays little, if any role in the aesthetics of Shaftesbury, Hutcheson, Burke, Kant, Schelling, Hegel etc., for instance. If one were to seek to retrieve the ancients’ lively appreciation of mathematical beauty, not just as impressionistic but as conceptually articulate, perhaps the most promising point of departure for a modern man would be Alexander Baumgarten, who in his pioneering Aesthetica of 1750/1758 systematically lays down general aesthetic principles, although to be sure he confines himself largely to poetics and does not dare to entertain the possibility of comprehending mathematics under his schema at all. The baleful hegemony of the analytic mode in the philosophy of mathematics has been just too stifling for anyone in modern times to have essayed anything of note in this direction.
Admonitory illustration: Michael Atiyah’s November 8, 2010 lecture at the Institute for Advanced Study in Princeton, available for viewing here. If one takes the trouble to follow Atiyah’s rudimentary presentation, he will rather be surprised by what it does not contain than by what it does. Presented are a handful of examples taken at random without any design and remaining elementary (he does not quote any very deep results that most would consider to be beautiful, say the Grothendieck-Riemann-Roch theorem). In Kant’s terms, he proceeds rhapsodically rather than systematically. The philosophical and aesthetic level of his dilettantish reflections is minimal, at best. For all its technical proficiency, his mind lacks organizing power. In consequence, the lecture issues in no identifiable or memorable thesis one could attach to Atiyah’s name. Clearly, after a long and stellar career, the great mathematician’s brain is stocked with a plenitude of pertinent observations on the nature of beauty in mathematics, but he seems curiously unwilling to exert himself to collect his disorganized heap of recollections into an articulated theoretical system of aesthetics, all the more unfortunate because this means that they will, to all intents and purposes, be lost to posterity and will have to be laboriously reconstructed by every budding student of mathematics during his formative years in graduate school, if at all.
This recensionist’s own philosophy of mathematics derives from the guiding influence of Proclus, that greatest of the late-antique Neoplatonists and spirited opponent of Christianity. For Proclus, as he understands him after having had the chance more than to glimpse at his commentary on Euclid, mathematics is a contemplation of the eternal ideas which consists in an ascent from the empirical world back to the originating form, and thence farther still to that which is beyond all determinate circumscription and, indeed, beyond being itself [hyperousios]. Through the medium of these intelligibilia, we perceive the pouring forth of the radiance of the divine. In this context, Kant’s transcendental aesthetic is very stimulating to ponder because intuition, to those in the know, is more than just a seeing, but also a ‘making seeing’, which is essentially the meaning of the German verb ‘denken’, understood etymologically as a subjunctive ‘bewirkendes Zeitwort’ of ‘dünken = it appears to me’; hence ‘denken = would cause to appear’. Thus, intuition appears to involve, in some sense, an intellectual force, where by force is intended ‘Kraft’ in its original denotation, not just a certain vector in Newtonian mechanics. A mathematical proposition is somehow just a precipitate of intuition into an expressible form. Since anyone sharing this recensionist’s exploratory attitude ought to be quite willing to risk assuming the axiom of choice and to work within naïve set theory, most of the time, unless some recalcitrant puzzle be encountered, the proper and fruitful concept of what intuition is or may be has nothing to do with what conventionally goes under the heading of L.E.J. Brouwer’s falsely modest ‘intuitionism’, which states that the human mind is incapable of entertaining anything not constructible in a finite number of steps. A view such as that herein advanced tends, in typical Neoplatonic fashion, to emanationism, which, if true, would be a flaw. Man cannot know, after all, what or who God is, but the perspective on concept formation in mathematical intuition just sketched helps us to picture what he could be, distantly to be sure!
Perhaps the most distinguished and, for the educated man, most congenial representative of the ancient philosophical stance is the Neoplatonist Proclus (fifth century AD), whose commentary on the first book of Euclid’s Elements forms the subject of the present review. Glenn Morrow has done us a great service with his translation and front matter in the edition published by the Princeton University Press. Out of the profusion of stimulating ideas to be found in this major work, to stay within the limitations of space we must set aside a narrow investigation of its technical terminology, rewarding as this would be, and seek to abstract the essential tenor of Proclus’ system. What will appear most foreign to modern sensibilities is his striking moral motivation for the pursuit of mathematics:
We must cultivate that science of geometry which with each theorem lays the basis for a step upward and draws the soul to the higher world, instead of letting it descend among sensibles to satisfy the common needs of mortals and, in aiming at these, neglect to turn away hence. (p. 69)
How, in practice, does this ascent take place?
This, then, is what learning [mathesis] is, recollection of the eternal ideas in the soul; and this is why the study that especially brings us the recollection of these ideas is called the science concerned with learning [mathematike]. Its name thus makes clear what sort of function this science performs. It arouses our innate knowledge, awakens our intellect, purges our understanding, brings to light the concepts that belong essentially to us, takes away forgetfulness and ignorance that we have from birth, sets us free from the bonds of unreason; and all this by the favor of the god who is truly the patron of this science, who brings our intellectual endowments to light, fills everything with divine reason, moves our souls towards Nous, awakens us as it were from our heavy slumber, through our searching turns us back upon ourselves, through our birthpangs perfects us, and through the discovery of pure Nous leads us to the blessed life. (p. 38)
Hence, Proclus, in keeping with Plato’s inscription above the entrance to the Academy [‘Let no one ignorant of geometry enter here’], sees in the pursuit of mathematics an inherently educative function, as the following three quotations describe:
The Timaeus calls mathematical knowledge the way of education, since it has the same relation to knowledge of all things, or first philosophy, as education has to virtue. Education prepares the soul for a complete life through firmly grounded habits, and mathematics makes ready our understanding and our mental vision for turning towards that upper world….For the beauty and order of mathematical discourse, and the abiding and steadfast character of this science, bring us into contact with the intelligible world itself and establish us firmly in the company of things that are always fixed, always resplendent with divine beauty, and ever in the same relationship to one another. (pp. 17-18)
Being thus endowed and led towards perfection, mathematics reaches some of its results by analysis, others by synthesis, expounds some matters by division, others by definition, and some of its discoveries binds fast by demonstration, adapting these methods to its subjects and employing each of them for gaining insight into mediating ideas. Thus its analyses are under the control of dialectic, and its definitions, divisions, and demonstrations are of the same family and unfold in conformity with the way of mathematical understanding. It is reasonable, then, to say that dialectic is the capstone of the mathematical sciences….Nous, then, wraps up the developments of the dialectical methods, binds together from above all the discursiveness of mathematical reasoning, and is the perfect terminus of the upward journey and of the activity of knowing. (pp. 35-36)
This thinking in geometry occurs with the aid of the imagination. Its syntheses and divisions of the figures are imaginary; and its knowing, though on the way to understandable being, still does not reach it, since the understanding is looking at things outside itself. At the same time the understanding sees them by virtue of what it has within; and though employing projections of its ideas, it is moved by itself to make them external. But if it should ever be able to roll up its extensions and figures and view their plurality as a unity without figure, then in turning back to itself it would obtain a superior vision of the partless, unextended, and essential geometrical ideas that constitute its equipment. This achievement would itself be the perfect culmination of geometrical inquiry, truly a gift of Hermes, leading geometry out of Calypso’s arms, so to speak, to more perfect intellectual insight and emancipating it from pictures projected in imagination. (pp. 44-45)
The following quotation illustrates well Proclus’ proclivity for descrying a moral significance in mathematical truths, thus anticipating the typological and symbolist mentality so characteristic of bookish medieval culture:
For these reasons, therefore, they refer right angles to the immaculate essences in the divine orders and their more particular potencies, as causes of the undeviating providence that presides over secondary things—for what is upright, uninclined to evil, and inflexible accords with the character of those high gods—whereas they say that obtuse and acute angles are left in the charge of the divinities that supervise the forthgoing of things and the change and variety of their powers. The obtuse angle is an image of the extension of the forms to everything, while the acute is a likeness of the cause that discriminates and activates all things….Rightly, then, they exhort the soul to make her descent into the world of generation after the undeviating form of the right angle, inclining no more to one side than to the other, nor being affected more by some things than by others, for the possession of fellow-feeling drags her down into the error and indeterminacy of matter. The perpendicular thus is also a symbol of directness, purity, undefiled unswerving force, and all such things, a symbol of divine and intelligent measure….Hence they say that virtue is like rightness, whereas vice is constituted after the fashion of the indeterminate obtuse and acute, possessing both excesses and deficiencies and showing by this more-and-less its own lack of measure. We shall therefore lay it down that the right among rectilinear angles is the image of perfection, undeviating energy, intelligent limit and boundary, and everything similar to them, and that the obtuse and acute angles are likenesses of indefinite change, irrelevant progression, differentiation, partition, and unlimitedness in general. (pp. 106-107)
Hugh of Saint-Victor’s De sacramentis would be an ideal place to draw out the inner connection between Proclus and the medieval sacramental world-view, but that must be deferred to another occasion. For the time being, let us round out our meditation by citing Proclus’ admirable and moving account of beauty:
To those who say these things we can reply by exhibiting the beauty of mathematics on the principles by which Aristotle attempts to persuade us. Three things, he says, are especially conducive to beauty of body or soul: order, symmetry, and definiteness. Ugliness in the body arises from the ascendancy of disorder and from a lack of shapeliness, symmetry, and outline in the material part of our composite nature; ugliness of mind comes from unreason, moving in an irregular and disorderly fashion, out of harmony with reason and unwilling to accept the principles it imposes; beauty, therefore, will reside in the opposites of these, namely, order, symmetry, and definiteness. These characters we find preeminently in mathematical science. We see order in its procedure of explaining the derivative and more complex theorems from the primary and simpler ones; for in mathematics later propositions are always dependent on the predecessors, and some are counted as starting-points, others as deductions from the primary hypotheses. We see symmetry in the accord of the demonstrations with one another and in their common reference back to Nous; for the measure common to all parts of the science is Nous, from which it gets its principles and to which it directs the minds of its students. And we see definiteness in the fixity and certainty of its ideas; for the objects of mathematical knowledge do not appear now in one guise and now in another, like the objects of perception or opinion, but always present themselves as the same, made definite by intelligible forms. If, then, these are the factors especially productive of beauty, and mathematics is characterized by them, it is clear that there is beauty in it. How could it be otherwise when Nous illumines this science from above and its earnest endeavor is to spur us to move from the sense world into that intelligible region? (pp. 22-23)
Certainly, contemporary practicing mathematicians have and wield an informal concept of beauty or elegance, but, as an index of ever-declining educational standards, it seems quite possible indeed that nobody in the past millenium and even longer (since Boethius and Scotus Eriugena, say) has devoted any rigorous reflection to what mathematical beauty truly consists in. It plays little, if any role in the aesthetics of Shaftesbury, Hutcheson, Burke, Kant, Schelling, Hegel etc., for instance. If one were to seek to retrieve the ancients’ lively appreciation of mathematical beauty, not just as impressionistic but as conceptually articulate, perhaps the most promising point of departure for a modern man would be Alexander Baumgarten, who in his pioneering Aesthetica of 1750/1758 systematically lays down general aesthetic principles, although to be sure he confines himself largely to poetics and does not dare to entertain the possibility of comprehending mathematics under his schema at all. The baleful hegemony of the analytic mode in the philosophy of mathematics has been just too stifling for anyone in modern times to have essayed anything of note in this direction.
Admonitory illustration: Michael Atiyah’s November 8, 2010 lecture at the Institute for Advanced Study in Princeton, available for viewing here. If one takes the trouble to follow Atiyah’s rudimentary presentation, he will rather be surprised by what it does not contain than by what it does. Presented are a handful of examples taken at random without any design and remaining elementary (he does not quote any very deep results that most would consider to be beautiful, say the Grothendieck-Riemann-Roch theorem). In Kant’s terms, he proceeds rhapsodically rather than systematically. The philosophical and aesthetic level of his dilettantish reflections is minimal, at best. For all its technical proficiency, his mind lacks organizing power. In consequence, the lecture issues in no identifiable or memorable thesis one could attach to Atiyah’s name. Clearly, after a long and stellar career, the great mathematician’s brain is stocked with a plenitude of pertinent observations on the nature of beauty in mathematics, but he seems curiously unwilling to exert himself to collect his disorganized heap of recollections into an articulated theoretical system of aesthetics, all the more unfortunate because this means that they will, to all intents and purposes, be lost to posterity and will have to be laboriously reconstructed by every budding student of mathematics during his formative years in graduate school, if at all.
This recensionist’s own philosophy of mathematics derives from the guiding influence of Proclus, that greatest of the late-antique Neoplatonists and spirited opponent of Christianity. For Proclus, as he understands him after having had the chance more than to glimpse at his commentary on Euclid, mathematics is a contemplation of the eternal ideas which consists in an ascent from the empirical world back to the originating form, and thence farther still to that which is beyond all determinate circumscription and, indeed, beyond being itself [hyperousios]. Through the medium of these intelligibilia, we perceive the pouring forth of the radiance of the divine. In this context, Kant’s transcendental aesthetic is very stimulating to ponder because intuition, to those in the know, is more than just a seeing, but also a ‘making seeing’, which is essentially the meaning of the German verb ‘denken’, understood etymologically as a subjunctive ‘bewirkendes Zeitwort’ of ‘dünken = it appears to me’; hence ‘denken = would cause to appear’. Thus, intuition appears to involve, in some sense, an intellectual force, where by force is intended ‘Kraft’ in its original denotation, not just a certain vector in Newtonian mechanics. A mathematical proposition is somehow just a precipitate of intuition into an expressible form. Since anyone sharing this recensionist’s exploratory attitude ought to be quite willing to risk assuming the axiom of choice and to work within naïve set theory, most of the time, unless some recalcitrant puzzle be encountered, the proper and fruitful concept of what intuition is or may be has nothing to do with what conventionally goes under the heading of L.E.J. Brouwer’s falsely modest ‘intuitionism’, which states that the human mind is incapable of entertaining anything not constructible in a finite number of steps. A view such as that herein advanced tends, in typical Neoplatonic fashion, to emanationism, which, if true, would be a flaw. Man cannot know, after all, what or who God is, but the perspective on concept formation in mathematical intuition just sketched helps us to picture what he could be, distantly to be sure!
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